Topic 22 Analysis of Variance

Transcription

1 Topic 22 Analysis of Variance Comparing Multiple Populations 1 / 14

2 Outline Overview One Way Analysis of Variance Sample Means Sums of Squares The F Statistic Confidence Intervals 2 / 14

3 Overview Two-sample t procedures are designed to compare the means of two populations. Our next step is to compare the means of several populations. To explain the methodology, we consider the data set gathered from the forests in Borneo. Figure: Danum Valley Conservation Area 3 / 14

4 Overview Example. The data on 30 forest plots in Borneo are the number of trees per plot. never logged logged 1 year ago logged 8 years ago n j ȳ j s i We compute these statistics from the data y 11,... y 1n1, y 21,... y 2n2 and y 31,... y 2n3, ȳ j = 1 n j n j i=1 y ij and s 2 j = 1 n j 1 n j (y ij ȳ i ) 2. i=1 4 / 14

5 Overview The basic question is: Are these means the same (the null hypothesis) or not (the alternative hypothesis)? The basic idea of the test is to examine the ratio of sbetween 2, the variance between groups (indicated by the variation in the center lines of the boxes) and sresidual 2, a statistic that measures the variances within groups. If the resulting ratio test statistic is sufficiently large, then we say, based on the data, that the means of these groups are distinct and we reject H never logged logged 1 year ago logged 8 years ago Figure: Side-by-side boxplots of the number of trees per plot. 5 / 14

6 One Way Analysis of Variance The hypothesis for one way analysis of variance is H 0 : µ j = µ k for all j, k and H 1 : µ j µ k for some j, k. The data {y ij, 1 i n j, 1 j q} represents that we have n j observation for the j-th group and that we have q groups. The total number of observations is denoted by n = n n q. The model is y ij = µ j + ɛ ij where ɛ ij are independent N(0, σ) random variables with σ 2 unknown. This allows us to define the likelihood and to use that to determine the analysis of variance F test as a likelihood ratio test. NB The model for analysis requires a common value σ for all of the observations. 6 / 14

7 Sample Means To set up the F statistic, we introduce two types of sample means: The within group means is the sample mean inside each of the groups, n j y j = 1 y ij, j = 1...., q. n j The mean of the data taken as a whole, known as the grand mean, i=1 y = 1 n n q j y ij = 1 n j=1 i=1 q n j ȳ j, j=1 the weighted average of the ȳ j with weights n j, the sample size in each group. Exercise. For the Borneo rain forest example, show that the grand mean is / 14

8 Sums of Squares Analysis of variance uses the total sums of squares SS total = n q j (y ij y) 2, j=1 i=1 the total square variation of individual observations from their grand mean. The test statistic is determined by decomposing SS total. We first rewrite the interior sum as n j (y ij y) 2 = (y ij y j ) 2 + n j (y j y) 2 = (n j 1)sj 2 + n j (y j y) 2. i=1 n j i=1 Here, s 2 j is the unbiased sample variance based on the observations in the j-th group. Exercise. Show the first equality above. (Hint: Begin with the difference in the two sums.) 8 / 14

9 Sums of Squares Summing the expression above over the groups j to yield the decomposition of the variation SS total = SS residual + SS between with SS residual = n q j (y ij y j ) 2 = j=1 i=1 q q (n j 1)sj 2 and SSbetween 2 = n j (y j y) 2. j=1 For the rain forest example, we find that and SS residual = (12 1) (12 1) (9 1) = SS between = 12 ( y) ( y) ( y) 2 ) = j=1 9 / 14

10 Sums of Squares source of degrees of sums of mean variation freedom squares square between groups q 1 SS between sbetween 2 = SS between/(q 1) residuals n q SS residual s 2 residual = SS residual /(n q) total n 1 SS total The q 1 degrees of freedom between groups is derived from the q groups minus 1 degree of freedom used to compute y. The n q degrees of freedom within the groups is derived from the n j 1 degree of freedom used to compute the variances s 2 j. 10 / 14

11 Sums of Squares The analysis of variance information for the Borneo rain forest data is summarized in the table below. source of degrees of sums of mean variation freedom squares square between groups residuals total / 14

12 The F Statistic The test statistic is F = s2 between s 2 residual = SS between/(q 1) SS residual /(n q). Under the null hypothesis, F is a constant multiple of the ratio of two independent χ 2 random variables, namely SS between and SS residual. This ratio is called an F random variable with q 1 numerator degrees of freedom and n q denominator degrees of freedom and written F q 1,n q 12 / 14

13 . For the rain forest data, F = s2 between s 2 residual = = The critical value for an 0.01 level test is So, we reject H 0 stating mean number of trees does not depend on logging history. > 1-pf(11.43,2,30) [1] > qf(0.99,2,30) [1] F Statistic df(x, 2, 30) x Figure: Upper tail critical values. The density for an F 2,30 random variable. The indicated values 3.316, 4.470, and are critical values for significance levels α = 0.05, 0.02, and 0.01, respectively. Exercise. Use R to determine these critical values. 13 / 14

14 Confidence Intervals Confidence intervals are determined using the data from all of the groups as an unbiased estimate s 2 residuals = SS residuals/(n q) for the variance, σ 2. This allows us to increase the degrees of freedom in the t distribution and reduce the margin of error. Thus, the γ-level confidence interval for µ j is ȳ j ± t (1 γ)/2,n q s residual / n j. The interval for the difference in µ j µ k is similar to that for a pooled two-sample t confidence interval, 1 ȳ j ȳ k ± t (1 γ)/2,n q s residual + 1. n j n k The 95% confidence interval for mean number of trees on a lot logged 1 year ago ± = ± = (9.369, ). 12 Exercise. Give the 95% confidence interval for the difference in trees between plots never logged plots versus logged 8 years ago. 14 / 14